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Evaluation of optimal educational model related to folk music in China with fuzzy preference relations.

1. Introduction

Now, music cultural diversity has become a mainstream trend in the existing social music culture development. Folk music is an important aspect of culture diversity. To inherit folk music is a good way to develop and improve culture diversity. More importantly, folk music as a kind of soft power in a country has attracted much attention all over the world (Kons et al., 2016). In China, folk music has a long history and cultural product of thousands of years. Thus, how to inherit and develop folk music is a big challenge that our country should face. For this purpose, education is a basic way to complete it. The educational value of folk music is defined as the positive and beneficial impact of folk music as an objective educational resource of society on individuals and society. The literature review of folk music especially the education of folk music will be demonstrated in the Section 2. In the modern society, Hereinto, there are many educational model of folk music in the existing education field. For example, on-line learning, face-to-face learning, telephone learning, class learning and so on has been applied in different district for different purpose. On-line learning is more flexible but lacks of good atmosphere. Face-to-face learning can provide better atmosphere but efficiency is poor. So, to evaluate different educational model to select an optimal educational model is a key task in the development of education of folk music. Because this evaluation is handled by considering several attributes and inviting several experts, it can be considered as a multiple attribute group decision making problem (Liao et al., 2016; Zhu and Xu, 2014).

To the best of our knowledge, the multiple attribute group decision making methods can be divided into two categories: direct assessments and preference relations (Li and Chen, 2016; Kim and Ahn, 1999). Here, it is easier for experts to provide preference relations (Kim et al., 1999). The commonly used valued preference relations are multiplicative preference relations and fuzzy preference relations (Sun and Ma, 2015; Wu and Chiclana, 2014). So, fuzzy preference relations as a basic and popular way are introduced in this paper to express the preference of experts. Taking the experts' characteristics with regard to knowledge, skills, experience and personality into account, they can choose their favourable preference relations to represent preferences. In fuzzy preference relations, two aspects need to be handled: consistency and consensus measure, and aggregation process. Consistency is associated with transitivity property. Consensus can guarantee the effectiveness of group decision making. The purpose of aggregation process is to aggregate preference relations of experts and further to generate the optimal educational model in this paper.

The rest of the paper is organized as follows: Section 2 reviews the literatures related to folk music and its education in China. Section 3 shows the proposed method including the construction of preference relation matrix, the definition of consistency measure and consensus measure and the aggregation process. Section 4 uses the proposed method to select an optimal educational model. Section 5 concludes this paper.

2. The existing studies of folk music education in China

Folk music is generated from the folk, spreads in the folk and reflects the history, society, labor, local conditions and customs and daily life of a nation. In China, folk music can be used in a broad and a narrow sense. From the perspective of broad sense, Chinese folk music includes Han nationality and other minority nationalities. On the other hand, from the perspective of narrow sense, Chinese folk music only includes Han nationality. The value of folk music is huge, which is considered to be related to a kind of soft power in a country and a typical representative of culture in a country.

In China, many researchers have devoted into studying folk music especially the education of folk music. Zhang (2015) rethought the influence and spread of folk music in minority nationalities. Zhu (2013) studied the development of folk music based on the perspective of soft power of culture. Yang emphasized the importance of education in folk music. Xie (2011) pointed out the education of folk music in the age of Internet. Internationally, many researchers also have discussed the value of music especially the education of music. Hietanen et al. (2016) focused on connecting student teachers' formal and autonomous learning to solidify their music education paths. Rauduvaite and Lasauskiene (2015) discussed opportunities of music education improvement integrating popular music and using innovative methods of personal meaning and emotional limitation in the context of theories of pedagogical thought. Semradova and Hubackova focused on characteristic of education, including distance education and comparison of the extent of teacher responsibility, which provides us a new way to think the education of folk music.

In brief, the education of folk music is important in a country. Thus, how to select an optimal educational model is a problem need to be solved as mentioned in Introduction.

3. The proposed method with fuzzy preference relation

As mentioned in Introduction, the group decision making method is proposed with fuzzy preference relations in this section. Details are demonstrated below.

3.1. he model of the group decision making

In a group decision making problem, several experts [e.sub.t] (t=1, 2, ..., q) are first invited to provide their preference using fuzzy preference relations for all the assessments [a.sub.ij] (i=i, 2, ..., m; j=i, 2, ..., n) to construct several decision matrixes [D.sup.t.sub.j] =[[[a.sub.ij]].sub.mxm] . For each alternative [A.sub.i] (i=i, 2, ..., m), any expert is required to express his/her preference on each attribute [C.sub.j] (j=i, 2, ..., n) which is constructed by the specific problem. Then, all experts specify the attribute weights of the n attributes denoted as w=[([w.sub.1], [w.sub.2], ..., [w.sub.n]).sup.T] with 0 [less than or equal to] [w.sub.j] [less than or equal to] 1 j=i, 2, ..., n) and [[SIGMA].sup.n.sub.i=1] [w.sub.j] =1 (Xu, 2014)

Therefore, q x n decision matrixes can be generated as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

3.2. fuzzy preference relation

Definition 1 (Herrera-Viedma and Porcel, 2009). Given a set of alternatives X={[x.sub.1], [x.sub.2], ..., [x.sub.m]}, a MPR matrix on X is defined as A [subset] XxX, A=[([a.sub.ij]).sub.mxm] , where [a.sub.ij] denotes the ratio of the preference intensity of alternative xi to that of [x.sub.j], which means [x.sub.i] is [a.sub.ij] times as good as [x.sub.j]. As suggested by Saaty (1980), the scale 1-9 is used to measure [a.sub.ij], where [a.sub.ij] = 1 indicates indifference between [x.sub.i] and [x.sub.j], [a.sub.ij] = 9 indicates [x.sub.i] is absolutely preferred to [x.sub.j], and [a.sub.ij] [member of] {1/8, ..., 1/2, 2, ..., 8} indicates the intermediate preference intensity between [x.sub.i] and [x.sub.j]. A multiplicative reciprocal A satisfies [a.sub.ij] x [a.sub.ji] = 1, [for all] i, j[member of]{1, ..., m}.

Similar to Definition 1, the concept of fuzzy preference relation should be defined in the following.

Definition 2. A fuzzy preference relation R on a set X = {[x.sub.1], [x.sub.2], ..., [x.sub.m]} is a fuzzy set on the product set X x X, i.e., it is characterized by a membership function [u.sub.R]: X x X [right arrow] [0, 1].

From the Definition 1, it can be deduced that a fuzzy preference relation R on X can be conveniently expressed by an mxm matrix R=[([r.sub.ij]).sub.mxm], where [r.sub.ij] = [u.sub.R]([x.sub.i], [x.sub.j]) (i, j=1, 2, ..., m) is interpreted as the preference degree or intensity of the alternative [x.sub.i] over [x.sub.j]. When [r.sub.ij] = 0.5, it indicates indifference between [x.sub.i] and [x.sub.j] ([x.sub.i] ~ [x.sub.j]); When [r.sub.ij] > 0.5, it indicates that [x.sub.i] is preferred to [x.sub.j] ([x.sub.i] [??] [x.sub.j]). In general, R = [([r.sub.ij]).sub.mxm] satisfies the additive reciprocity property, namely, [r.sub.ij] + [r.sub.ji] = 1 for all i, j = 1, 2, ..., m. Without loss of generality, in this paper, R is assumed to be additive reciprocal.

Then, in a group decision making, consistency is one of the important factors (HerreraViedma et al., 2007). In this paper, the concept of additive consistency is defined below.

Definition 3. The fuzzy preference relation R = [([r.sub.ij]).sub.mxm] is additively consistent, if it satisfies

[r.sub.ij] = [r.sub.ik] + [r.sub.kj] - 0.5 (2)

for all i, j, k = 1, 2, ..., m with i < k < j and [r.sub.ij] + [r.sub.ji] = 1.

According to the Eq. (1), Herrera-Videma classified three cases which are denoted by

1. [r.sup.k,1.sub.ij] = [r.sub.ik] + [r.sub.kj] - 0.5 by [r.sub.ij] = [r.sub.ik] + [r.sub.kj] - 0.5;

2. [r.sup.k,2.sub.ij] = [r.sub.kj] + [r.sub.ki] + 0.5 by [r.sub.kj] = [r.sub.ki] + [r.sub.ij] - 0.5;

3. [r.sup.k,3.sub.ij] = [r.sub.ki] + [r.sub.jk] + 0.5 by [r.sub.ik] = [r.sub.ij] + [r.sub.jk] - 0.5.

Thus, the three deviations can be deduced from the mentioned three cases.

1. [epsilon][r.sup.k,1.sub.ij] = [n.summation over (k=1,k[not equal to]i,j)] [absolute value of [r.sup.k,1.sub.ij] - [r.sub.ij]]/n-2;

2. [epsilon][r.sup.k,2.sub.ij] = [n.summation over (k=1,k[not equal to]i,j)] [absolute value of [r.sup.k,2.sub.ij] - [r.sub.ij]]/n-2;

3. [epsilon][r.sup.k,3.sub.ij] = [n.summation over (k=1,k[not equal to]i,j)] [absolute value of [r.sup.k,3.sub.ij] - [r.sub.ij]]/n-2.

3 n - 2

Then, using the deviations the consistency level related to the fuzzy preference relation [r.sub.ij] (i < j) can be defined in the following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The lower the [CL.sub.rij] is, the more inconsistent the fuzzy preference value [r.sub.ij] is. When there is no inconsistency at all, [CL.sub.rij] =1,

Based on the Eq. (3), the consistency level of a fuzzy preference relation R=[([r.sub.ij]).sub.mxm] can be proposed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Here, [CL.sub.R] [member of] [0,1].

3.3. group consensus reaching process

Consistency is a critical issue in decision making with preference relations because the lack of consistency in preference relations may lead to unreasonable results. In general, the consistency of a preference relation is identified by transitivity properties among preference judgments. There are different types of transitivity properties including the weak transitivity, the max-min transitivity, the max-max transitivity, the restricted maxmin transitivity, the restricted max-max transitivity and the multiplicative transitivity.

When a fuzzy preference relation R=(rij) mxm is complete, one can easily obtain that the fuzzy preference relation R = [([[??].sub.ij]).sub.mxm] is additively consistent, where

[[??].sub.ij] = [n.summation over (k=1)] [r.sub.ik] + [r.sub.kj]/n 0.5, i, j = 1, 2, ..., n. (5)

Definition 4. Let R = [([r.sub.ij]).sub.mxm] be a fuzzy preference relation, and [??] = [([[??].sub.ij]).sub.mxm] be the associated additively consistent fuzzy preference relation. The additive consistency index (ACI) of R is defined as follows:

ACI(R)- 1- 2/n(n-1) [n.summation over (i,j=1,i<j)]d([r.sub.ij],[[??].sub.ij]) (6)

Here d([r.sub.ij],[[??].sub.ij]) = [absolute value of [r.sub.ij] - [[??].sub.ij], i, j = 1 2 ..., n.

Consensus is defined as general or widespread agreement of the experts. In order to reach group consensus, the first thing we should do is to measure the closeness among the expert's opinions. As full and unanimous consensus is often unreachable, an alternative approach is to use softer consensus measure.

Definition 5. Let [R.sup.t] = [([r.sup.t.sub.ij]).sub.mxm] be a fuzzy preference relation provided by the expert [e.sub.t](t=1, 2, ..., q), and R* = [([r.sup.*.sub.ij]).sub.mxm] be the associated collective fuzzy preference relation.

The group consensus index (GCI) of [R.sup.t] is defined as follows:

GCI([R.sup.t]) = 1 - 2/n(n-1) [n.summation over (i,j=1,i<j)] d([r.sup.t.sub.ij], [r.sup.*.sub.ij]) (7)

Given a critical value denoted by [phi]. It can be used to measure the consensus in a group decision making. That is, if GCI([R.sup.t])>[phi], the group consensus can be considered to be reached. In this paper, average opinion of experts can be regarded as the value of [phi].

[phi] = [q.summation over (t)] [w.sub.t]GCI ([R.sup.t]), (8)

where [w.sub.t] denotes the weight of the expert t.

3.4.The process of the decision making method

In this sub-section, the process based on group consensus measure and consistency measure is demonstrated in the following.

Step 1. Form a MAGDA problem. A manager selects the experts, identifies attributes and their types (benefit and cost), and lists alternatives to form a MAGDM problem.

Step 2. Prepare for the group consensus in order to solve the MAGDA problem. The manager identifies MAXCYCLE, the maximal number of times of GAD to avoid the delayed convergence of collective solution, sets CYCLE = 0, a cycle counter; decides relative weights of experts.

Step 3. Collect experts' FPRs. All experts independently give their FPRs on all attributes of all alternatives.

Step 4. Decide whether the group consensus is reached. It is determined whether the group consensus is reached using Eqs. (7)-(8). If so, go to Step 8. Otherwise, go to Step 5.

Step 5. Organize the GAD and collect the renewed FPRs of experts. If CYCLE > MAXCYCLE then go to Step 8. Otherwise, all experts renew their FPRs. Furthermore, CYCLE = CYCLE +1 is set by the manager, and the GAD is organized by the manager to help the experts to renew their FPRs. After that, go to Step 6.

Step 6. Form the aggregated group FPRs on each alternative. Calculate the aggregated group FPRs on each alternative.

Step 7. Generate a ranking order of alternatives.

Step 8. Finish the procedure. The manager checks whether CYCLE > MAXCYCLE holds. If so, a conclusion of no group consensus based solution for the MAGDA problem can be drawn. Otherwise, the optimum alternative or the ranking order of alternatives can be selected as a group consensus based solution to the MAGDA problem reaching the predefined group consensus.

4. Illustrative example

Folk music as a culture can guarantee the stability of society in our country. It is widely accepted that music can connect everyone all over the world beyond the geographical limitations. Hereinto, the education of folk music is critical to develop folk music. Thus, we should propose a scientific method to select an optimal educational model of folk music for different purposes.

Using the proposed method in Section 3, three experts are invited firstly to provide their preference to three alternative educational models including telephone, Internet and class denoted by [A.sub.1], [A.sub.2] and [A.sub.3]. In addition, three attributes are specified by the experts including quality, cost, benefits denoted by [C.sub.1], [C.sub.2], and [C.sub.3] .

The experts provide fuzzy preference relations of each alternative on each attribute, respectively. Therefore, 9 decision matrixes can be constructed. Here, the attribute weight can be determined as w = [(0.3, 0.4, 0.3).sup.T] by the experts. Because of the limitation of content, we select three decision matrixes in the following to demonstrate the preference of the experts.

Then, based on the Tables 1-3 and omitted 6 Tables, fuzzy preference relations on each attribute related to each alternative can be aggregated into collective fuzzy preference relations between pairs of alternatives in the Table 4.

Finally, the ranking order is demonstrated as [A.sub.2] [??] [A.sub.3] [??] [A.sub.1] which indicates that [A.sub.2] is the optimum educational model in Table 5.

5. Conclusion

Folk music as an aspect of culture diversity and soft power in a country has been attracted much attention. In China, folk music has a long history and cultural product of thousands of years. Thus, how to inherit and develop folk music is a big challenge that our country should face. In order to inherit folk music to the next generation, different educational models are generated. Then, for different purpose, an appropriate educational model should be selected. Therefore, compared to other direct assessments, fuzzy preference relation is introduced in this paper to objectively express the preference of experts. In fuzzy preference relations, two aspects need to be handled: consistency and consensus measure, and aggregation process. Consistency is associated with transitivity property. Consensus can guarantee the effectiveness of group decision making. The purpose of aggregation process is to aggregate preference relations of experts and further to generate the optimal educational model in this paper.

Recebido/Submission: 11/05/2016

Aceitacao/Acceptance: 18/07/2016

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Rong Chen (1,2)

chenrongmusicology@163.com

(1) School of Public Administration, Hohai University, Nanjing, Jiangsu 210098, China

(2) Hubei University of Education, Wuhan, Hubei 430205, China

Table 1 - Decision matrix with fuzzy preference relation
provided by expert el related to attribute [C.sub.1]

[C.sub.1]    [A.sub.1]   [A.sub.2]   [A.sub.3]

[A.sub.1]       0.5       0.4345      0.4835
[A.sub.2]     0.5655        0.5       0.6736
[A.sub.3]     0.5155      0.3264        0.5

Table 2 - Decision matrix with fuzzy preference relation
provided by expert e2 related to attribute [C.sub.2]

[C.sub.2]   [A.sub.1]   [A.sub.2]   [A.sub.3]

[A.sub.1]      0.5       0.3575      0.5038
[A.sub.2]    0.6425        0.5       0.6886
[A.sub.3]    0.4962      0.3114        0.5

Table 3 - Decision matrix with fuzzy preference relation
provided by expert e2 related to attribute [C.sub.3]

[C.sub.1]   [A.sub.1]   [A.sub.2]   [A.sub.3]

[A.sub.1]      0.5       0.4749      0.4673
[A.sub.2]    0.5251        0.5       0.4987
[A.sub.3]    0.5327      0.5013        0.5

Table 4 - Collective decision matrix with fuzzy preference relation

[C.sub.1]   [A.sub.1]   [A.sub.2]   [A.sub.3]

[A.sub.1]      0.5       0.4178      0.4867
[A.sub.2]    0.5822        0.5       0.5951
[A.sub.3]    0.5133      0.4049        0.5

Table 5 - The ranking order of alternatives

[C.sub.1]    Ranking-order

[A.sub.1]          3
[A.sub.2]          1
[A.sub.3]          2
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